Faber–Krahn inequalities in sharp quantitative form

Abstract

The classical Faber–Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet Laplacian among sets with given volume. In this article we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and by Bhattacharya and Weitsman. More generally, the result applies to every optimal Poincaré–Sobolev constant for the embeddings $W_0^{1,2}(\Omega )\hookrightarrow L^q(\Omega )$.